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Problem Set 2 — Graphical Analysis
1. Use graphical analysis to describe all orbits of the functions below. Also draw their
phase portraits.
(a)
(b)
(c)
(d)
(e)
F (x) = 2x
F (x) = 1 − 2x
F (x) = x2
F (x) = x − x2
F (x) = sin(x)
2. Use graphical analysis to find all the points whose orbits tend to infinity, i.e.
{x0 | F n (x0 ) → ±∞}, for the following functions:
(a) F (x) = x2 + 1
(
2x
0 ≤ x ≤ 1/2
(b) F (x) =
2 − 2x 1/2 < x ≤ 1
3. Completely analyse the orbits of the following functions:
(a)
(b)
(c)
(d)
F (x) = 12 x − 2
F (x) = |x|
F (x) = −x5
F (x) = ex
4. Analyse the orbits of the function F (x) = |x − 2|. Draw different types of orbits in
different colours. You will be able to find fixed points, eventually fixed points, periodic
points and eventually periodic points.
5. Let F (x) = x2 − 56 . Find the fixed point(s) of F . Using the fixed point(s) (or otherwise)
find the cycle of prime period 2.
6. Let F (x) = ax+b. Answer the following questions about the dynamics of F for various
values of a and b:
(a)
(b)
(c)
(d)
(e)
Find the fixed points of F .
For what values of a and b does F have no fixed points?
For what values of a and b does F have infinitely many fixed points?
For which values of a and b does F have exactly one fixed point?
If F has exactly one fixed point and |a| < 1, what is the behaviour of all obits
under F ? Use graphical analysis.
(f) Similarly, if |a| > 1 what is the behaviour of all orbits under F ?
(g) If a = 1 describe the orbits of F for b < 0, b = 0 and b > 0?
(h) Similarly, if a = −1 describe the orbits of F for b < 0, b = 0 and b > 0?
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